Unconditionally energy stable numerical schemes for hydrodynamics

University of South Carolina Scholar Commons Theses and Dissertations 2017 Unconditionally Energy Stable Numerical Schemes for Hydrodynamics Coupled Fluids Systems Alexander Yuryevich Brylev University of South Carolina Follow this and additional works at: https://scholarcommons.edu/etd Part of the Mathematics Commons Recommended Citation Brylev, A. Unconditionally Energy Stable Numerical Schemes for Hydrodynamics Coupled Fluids Systems. Retrieved from https://scholarcommons.edu/etd/4010 This Open Access Dissertation is brought to you by Scholar Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of Scholar Commons.

For more information, please contact digres@mailbox. Unconditionally Energy Stable Numerical Schemes for Hydrodynamics Coupled Fluids Systems by Alexander Yuryevich Brylev Bachelor of Arts Hamline University 2006 Master of Science New Mexico State University 2011 Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics College of Arts and Sciences University of South Carolina 2017 Accepted by: Xiaofeng Yang, Major Professor Lili Ju, Committee Member Zhu Wang, Committee Member Xinfeng Liu, Committee Member Dewei Wang, Outside Committee Member Cheryl L. Addy, Vice Provost and Dean of the Graduate School Acknowledgments First of all, I would like to thank my thesis director, professor Xiaofeng Yang, for his support and guidance in writing this thesis. He held numerous office hours to work with me on it, was always prompt in answering e-mails and any questions I had about the research.

This work wouldn’t be possible without him. I also appreciate being funded by a research grant several times. Next, I thank all my professors during my first two years at the University of South Carolina for helping me build the foundations I needed to be able to succeed in my project. Courses in Computational Mathematics (MATH 708 and MATH 709), taught by professors Lili Ju and Xiaofeng Yang, as well as Numerical Differential Equations (MATH 726) and Applied Mathematics (MATH 720 and MATH 721), taught by professor Hong Wang, turned out to be particularly valuable.

Finally, I would like to thank the department of Mathematics at the University of South Carolina for accepting me on a PhD program and providing me financial support in the form of teaching assistanship. I really enjoyed my work experience and it was great to be a part of the Gamecock family! ii Abstract The thesis consists of two parts. In the first part we propose several second order in time, fully discrete, linear and nonlinear numerical schemes to solve the phase-field model of two-phase incompressible flows in the framework of finite ele- ment method. The schemes are based on the second order Crank-Nicolson method for time disretizations, projection method for Navier-Stokes equations, as well as several implicit-explicit treatments for phase-field equations.

The energy stability, solvability, and uniqueness for numerical solutions of proposed schemes are further proved. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed schemes thereafter. In the second part we consider the numerical approximations for the model of smectic-A liquid crystal flows. The model equation, that is derived from the varia- tional approach of the de Gennes energy, is a highly nonlinear system that couples the incompressible Navier-Stokes equations and two nonlinear coupled second-order elliptic equations.

Based on some subtle explicit-implicit treatments for nonlinear terms, we develop unconditionally energy stable, linear, decoupled time discretiza- tion scheme. We also rigorously prove that the proposed scheme obeys the energy dissipation law. Various numerical simulations are presented to demonstrate the ac- curacy and the stability thereafter. iii Table of Contents Acknowledgments.

iii List of Tables. v List of Figures. vi Chapter 1 Numerical analysis of certain schemes for phase field models of two-phase incompressible flows .2 The PDE System and Energy Law.3 Second Order, Semi-Discrete Schemes and Their Energy Stability.4 Fully Discrete Schemes and Energy Stability. 27 Chapter 2 Numerical approximations for smectic–A liquid crys- tal flows .2 The smectic-A liquid crystal fluid flow model and its energy law.

53 v List of Tables Table 1.1 Cauchy convergence test for the linear scheme (1.63) solv- ing ACNS system; errors are measured in L2 norm; 2k grid points in each direction for k from 4 to 8, δt = 0.2 Cauchy convergence test for the linear scheme (1.76) solv- ing CHNS system; errors are measured in L2 norm; 2k grid points in each direction for k from 4 to 8, δt = 0.3 Cauchy convergence test for the nonlinear convex-splitting scheme (1.81) solving ACNS system; errors are measured in L2 norm; 2k grid points in each direction for k from 4 to 8, δt = 0. 28 vi List of Figures Figure 1.1 Temporal evolution of a circular domain driven by mean cur- vature without hydrodynamic effects. The parameters are η = 1.2 The areas of the circle as a function of time. The slope of the line is −6.2842 and the theoretical slope is −2π.3 Snapshots of the relaxation of a square shape by the ACNS system.4 Zero contour plots of the merging and relaxation of two kissing circles by the CHNS system.

From left to right, t = 0.5 Filled contour plots in gray scale of the rising bubble by the CHNS system. From left to right, t = 0.1 The L2 errors of the layer funciton φ, the director field d = (d1 , d2 ), the velocity u = (u, v) and pressure p. The slopes show that the scheme is asymptotically first-order accurate in time.2 Snapshots of the layer function φ are taken at t = 0, 0.3 Snapshots of the director field d are taken at t = 0, 0.4 Time evolution of the free energy functional of Example 2.5 Snapshots of the layer function φ are taken at t = 0, 0.6 Snapshots of the director field d are taken at t = 0, 0.7 Snapshots of the profile for the first component u(y) of the velocity field u = (u, v) at the center (x = 2) and t = 0, 0. 52 viii Chapter 1 Numerical analysis of certain schemes for phase field models of two-phase incompressible flows 1.1 Introduction Interfacial problems have attracted much attention of scientists for over a century.

A classical approach to dealing with such problems was to introduce a mesh with grid points on the interfaces which deforms according to the motion of the boundary. This method, however, had a drawback that large displacement or deformation of internal domains could cause computational issues such as mesh entaglement. To overcome this, sophisticated remeshing schemes were often times used [57]. Other methods which proved to work well were the volume-of-fluid (VOF) [48, 49], the front-tracking [40, 41] and the level-set [61, 78] fixed-grid methods, where the interfacial tension is represented as a body-force or bulk-stress spreading over a narrow region covering the interface.

The VOF method is a numerical technique for tracking and locating the interface between the fluids using the marker function. The disadvatage of this method is in its difficulty maintaining the sharp interface between the fluids and the computation of the surface tension. The level-set method has improved the accuracy and, hence, the applicability of the VOF method. The problem with the level-set method occurs when one tries to use it in an advection field, for example, uniform or rotational velocity field.

In this case the shape and size of the level set must be conserved, however, the method does not guarantee this, so the level set may get significantly distorted and vanish over several time steps. This requires 1 the use of high-order finite difference schemes, such as high-order essentially non- oscillatory (ENO) schemes [47], and even then, the feasibility of long-time simulations is questionable. To overcome this difficulty, more sophisticated methods have been designed, such as combinations of the level set method with tracing marker particles advected by the velocity field [60]. In the front-tracking method a separate front marks the interface but a fixed grid, only modified near the front to make a grid line follow the interface, is used for the fuid within each phase.

Phase-field or diffuse-interface model is another mathematical model for solving various interfacial problems. In recent years it has been successfully used to simulate dynamical processes in many fields and has become one of the major tools to study various systems arising from the energy-based variational formalism. The method employs an order parameter, called the phase field, and substitutes boundary condi- tions at the interface by the partial differential equation involving this new variable. The phase field is assigned distinct values on each phase (for example, -1 and 1) and a thin smooth transition layer marking the interface is defined as the set of all points where the phase field takes a certain value (for example, 0).

Hence the dynamics of the interface can be simulated on a fixed grid without explicit interface tracking, which renders the diffuse interface method an attractive numerical approach to sim- ulate free moving/deforming interfacial problems. Based on variational approaches, the governing system can be derived from the total free energy, which usually leads to some well-posed nonlinear partial differential equations. This makes it possible to carry out mathematical analysis and design numerical schemes which preserve the thermo-dynamically consistent dissipation law (energy-stable) at the discrete level. The preservation of such laws is critical for numerical methods to capture the correct long time dynamics.

The dynamics of phase field models can be described by either the Allen-Cahn equation [4] or the Cahn-Hilliard equation [8, 9] based on choices of Sobolev spaces 2 of the variational approach. In details, the Allen-Cahn equation is a second-order equation, which is easier to solve numerically but does not conserve the volume frac- tion, while the Cahn-Hilliard equation is a fourth-order equation which conserves the volume fraction but is relatively harder to solve numerically. Basically, the coarse- graining (macroscopic) process described by these two equations may undergo rapid changes near the interface, so the noncompliance of energy dissipation laws may lead to spurious numerical solutions if the grid and time step sizes are not carefully controlled [52, 36]. Thus, from the numerical point of view, people are particularly in- terested in designing simple, efficient and energy stable numerical schemes satisfying discrete energy dissipations laws.

There are several challenges to construct the efficient numerical schemes to solve the hydrodynamics coupled phase field model numerically, namely, i) the small in- terfacial width introduces tremendous amount of stiffness into the system ii) the nonlinear coupling between the phase variable and velocity due to the nonlinear con- vections and stresses, iii) the coupling between the velocity and pressure in the fluid momentum equation. It is by no means an easy task, in particular, the development of any efficient and accurate numerical schemes while maintaining the dissipative energy law. It is remarkable that many attempts have been made in this direction recently (cf. a comprehensive summary in [65].

However, due to the complexity of the nonlinear convection terms and stresses in the system, most of developed schemes are either only first-order in time [44, 69, 67], or are nonlinear schemes which need some efficient iterative solvers [76, 15], or only focus on the no flow case [59, 77, 31], or unable to provide the stability analysis [17]. There are very few works with the focus on the development of the second order schemes for the hydrodynamics coupled phase field model. Recently, in [32], a second order, unconditionally stable, semi-discrete scheme for 3 the hydrodynamics coupled Cahn-Hilliard phase field model was developed, which could be regarded as one of the limited successful efforts in the development of second order schemes. However, in [32], first, the schemes for the computation of the phase field variable are nonlinear thanks to the application of the convex splitting approach, thus one in turn needs some efficient iterative solvers.

Second, the computation of the phase variable is always coupled with that of the velocity. Third, the proof of energy law is only for the time discretization case.